{"id":2275,"date":"2023-11-04T02:53:32","date_gmt":"2023-11-04T06:53:32","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2275"},"modified":"2024-10-25T12:12:15","modified_gmt":"2024-10-25T16:12:15","slug":"number-theory-challenge-1","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2275","title":{"rendered":"Number Theory Challenge &#8211; 1"},"content":{"rendered":"\n<p>Let $n=4k+3$ is a prime number and $a^2+b^2\\equiv 0\\pmod{n}$, prove that $a\\equiv b \\equiv 0\\pmod{n}$.<\/p>\n\n\n\n<button onclick=\"toggle_visibility('nt-chall-1-solution')\">Click for the solution<\/button>\n<div style=\"display:none\" id=\"nt-chall-1-solution\">\n\n\n\n<p><strong>Proof<\/strong>: If $a\\equiv 0\\pmod{n}$, because $a^2+b^2\\equiv 0\\pmod{n}$, $b\\equiv 0\\pmod{n}$.<\/p>\n\n\n\n<p>If $a\\not\\equiv 0\\pmod{n}$, because $n$ is a prime number, according to <a rel=\"noreferrer noopener\" href=\"https:\/\/en.wikipedia.org\/wiki\/Fermat%27s_little_theorem\" target=\"_blank\">Fermat&#8217;s Little Theorem<\/a>, we have $$a^{n-1}\\equiv 1\\pmod{n}\\tag{1}$$<\/p>\n\n\n\n<p>Because $a^2+b^2\\equiv 0\\pmod{n}$, we have $b\\not\\equiv 0\\pmod{n}$. Similarily, we $$b^{n-1}\\equiv 1\\pmod{n}\\tag{2}$$ Therefore, we have the following: $$a^{n-1}+b^{n-1}\\equiv 2\\pmod{n}\\tag{3}$$<\/p>\n\n\n\n<p>Because $n=4k+3$, and $a^2+b^2\\equiv 0\\pmod{n}$, we have $$a^{n-1}+b^{n-1}\\equiv a^{4k+2}+b^{4k+2}\\equiv (a^2)^{2k+1}+(b^2)^{2k+1}$$ $$\\equiv (a^2+b^2)\\sum_{i=0}^{2k}(-1)^{i}(a^2)^{2k-i}(b^2)^i\\equiv 0\\pmod{n}\\tag{4}$$<\/p>\n\n\n\n<p>Therefore the assumption of $a\\not\\equiv 0\\pmod{n}$ leads conflicting equation $(3)$ and $(4)$. So, we can only have $\\boxed{a\\equiv b\\equiv 0\\pmod{n}}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Let $n=4k+3$ is a prime number and $a^2+b^2\\equiv 0\\pmod{n}$, prove that $a\\equiv b \\equiv 0\\pmod{n}$. Click for the solution Proof: If $a\\equiv 0\\pmod{n}$, because $a^2+b^2\\equiv 0\\pmod{n}$, $b\\equiv 0\\pmod{n}$. If $a\\not\\equiv 0\\pmod{n}$, because $n$ is a prime number, according to Fermat&#8217;s &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2275\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[16],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2275"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2275"}],"version-history":[{"count":23,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2275\/revisions"}],"predecessor-version":[{"id":2311,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2275\/revisions\/2311"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2275"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2275"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}